Precoding codebook design for single user MIMO

ABSTRACT

A transmitter is for use with multiple transmit antennas and includes a precoder unit configured to precode data for a transmission using a precoding matrix selected from a codebook, wherein the codebook corresponds to the following three transmission properties for an uplink transmission: 1) all precoding elements from the precoding matrix have a same magnitude, 2) each precoding element from the precoding matrix is taken from a set of finite values and 3) there is only one non-zero element in any row of the precoding matrix. The transmitter also includes a transmit unit configured to transmit the precoded data.

CROSS-REFERENCE TO PROVISIONAL APPLICATION

This application claims the benefit of U.S. Provisional Application No.61/154,676, filed by Eko N. Onggosanusi, Badri Varadarajan, Runhua Chenand Zukang Shen on Feb. 23, 2009, entitled “Precoding Codebook DesignWith Low Power Rating” commonly assigned with this application andincorporated herein by reference.

This application also claims the benefit of U.S. Provisional ApplicationNo. 61/172,529, filed by Eko N. Onggosanusi, Badri Varadarajan, RunhuaChen and Zukang Shen on Apr. 24, 2009, entitled “Progressing On 4TxCodebook Design For Uplink SU-MIMO” commonly assigned with thisapplication and incorporated herein by reference.

This application additionally claims the benefit of U.S. ProvisionalApplication No. 61/181,474, filed by Eko N. Onggosanusi, BadriVaradarajan, Runhua Chen and Zukang Shen on May 27, 2009, entitled “4TxCodebook For Uplink SU-MIMO: Cubic Metric (CM) Preserving Design”commonly assigned with this application and incorporated herein byreference.

This application further claims the benefit of U.S. ProvisionalApplication No. 61/219,212, filed by Eko N. Onggosanusi, BadriVaradarajan, Runhua Chen, Zukang Shen and Tarik Muharemovic on Jun. 22,2009, entitled “4Tx Codebook For Uplink SU-MIMO: Cubic Metric (CM)Preserving Design” commonly assigned with this application andincorporated herein by reference.

This application yet further claims the benefit of U.S. ProvisionalApplication No. 61/219,230, filed by Eko N. Onggosanusi, BadriVaradarajan, Runhua Chen, Zukang Shen and Tarik Muharemovic on Jun. 22,2009, entitled “4Tx Codebook For Uplink SU-MIMO: Rank 3” commonlyassigned with this application and incorporated herein by reference.

This application still further claims the benefit of U.S. ProvisionalApplication No. 61/242,274, filed by Eko N. Onggosanusi, BadriVaradarajan, Runhua Chen, Zukang Shen and Tarik Muharemovic on Sep. 14,2009, entitled “4Tx Codebook For Uplink SU-MIMO: Cubic Metric (CM)Preserving Design” commonly assigned with this application andincorporated herein by reference.

This application additionally further claims the benefit of U.S.Provisional Application No. 61/301,909, filed by Eko N. Onggosanusi,Badri Varadarajan, Runhua Chen, Zukang Shen and Tarik Muharemovic onFeb. 5, 2009, entitled “4Tx Codebook For Uplink SU-MIMO: Cubic Metric(CM) Preserving Design” commonly assigned with this application andincorporated herein by reference.

TECHNICAL FIELD

The present disclosure is directed, in general, to a communicationsystem and, more specifically, to a transmitter, a receiver and methodsof operating a transmitter and a receiver.

BACKGROUND

Multiple-input, multiple-output (MIMO) communication systems providelarge increases in throughput due to their ability to support multiple,parallel data streams. These data streams are each transmitted fromdifferent spatial transmission layers, which employ either physical orvirtual antennas. Generally, MIMO transmissions employ one or moreparallel spatial streams that are forward error correction (FEC) encodedemploying codewords. Each stream or codeword is then mapped to one ormore transmission layers. Mapping of a single encoded stream to multiplelayers may be simply achieved by distributing the encoded stream to allavailable layers. That is, the serial stream from the FEC encoding isconverted to parallel streams on different layers. The number of spatialtransmission layers employed is called the rank of the transmission.

The number of transmission layers used by a transmission may be anynumber up to the total number of physical antennas available. A signalon the virtual antennas is typically converted to a signal on thephysical antennas by using linear precoding. Linear precoding consistsof linearly combining the virtual antenna signals to obtain the actualsignals to be transmitted. The effectiveness of a MIMO communication isdependent on the particular set of possible precoders (called acodebook) that may be used in the transmission. Although many currentprecoding codebooks exist, further improvements would prove beneficialin the art.

SUMMARY

Embodiments of the present disclosure provide a transmitter, a receiverand methods of operating a transmitter and a receiver. In oneembodiment, the transmitter is for use with multiple transmit antennasand includes a precoder unit configured to precode data for atransmission using a precoding matrix selected from a codebook, whereinthe codebook corresponds to the following three transmission propertiesfor an uplink transmission: 1) all precoding elements from the precodingmatrix have a same magnitude, 2) each precoding element from theprecoding matrix is taken from a set of finite values and 3) there isonly one non-zero element in any row of the precoding matrix. Thetransmitter also includes a transmit unit configured to transmit theprecoded data.

In another embodiment, the receiver includes a receive unit configuredto receive precoded data from a transmitter having multiple transmitantennas. The receiver also includes a precoder selection unitconfigured to select a precoding matrix from a codebook, wherein thecodebook corresponds to the following three transmission properties foran uplink transmission: 1) all precoding elements from the precodingmatrix have a same magnitude, 2) each precoding element from theprecoding matrix is taken from a set of finite values, and 3) there isonly one non-zero element in any row of the precoding matrix.

In another aspect, the present disclosure provides an embodiment of themethod operating a transmitter for use with multiple transmit antennas.The method includes selecting a precoding matrix from a codebook,wherein the codebook corresponds to the following three transmissionproperties for an uplink transmission: 1) all precoding elements fromthe precoding matrix have a same magnitude, 2) each precoding elementfrom the precoding matrix is taken from a set of finite values and 3)there is only one non-zero element in any row of the precoding matrix,and transmitting the precoded data.

In yet another aspect, the present disclosure also provides anembodiment of the method of operating a receiver for use with multipletransmit antennas. The method includes receiving precoded data from atransmitter having multiple transmit antennas, and selecting a precodingmatrix from a codebook, wherein the codebook corresponds to thefollowing three transmission properties for an uplink transmission: 1)all precoding elements from the precoding matrix have a same magnitude,2) each precoding element from the precoding matrix is taken from a setof finite values and 3) there is only one non-zero element in any row ofthe precoding matrix.

The foregoing has outlined preferred and alternative features of thepresent disclosure so that those skilled in the art may betterunderstand the detailed description of the disclosure that follows.Additional features of the disclosure will be described hereinafter thatform the subject of the claims of the disclosure. Those skilled in theart will appreciate that they can readily use the disclosed conceptionand specific embodiment as a basis for designing or modifying otherstructures for carrying out the same purposes of the present disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present disclosure, referenceis now made to the following descriptions taken in conjunction with theaccompanying drawings, in which:

FIG. 1 illustrates a transmitter 100 as provided by one embodiment ofthe disclosure;

FIG. 2 illustrates a receiver 200 as provided by a correspondingembodiment of the disclosure;

FIGS. 3A, 3B and 3C illustrate an example of transmitter elementindexing for a transmitter and a receiver such as the transmitter 100and receiver 200 shown in FIGS. 1 and 2;

FIGS. 4A, 4B, 4C and 4D illustrate examples of performance comparisonsfor Rank-1 codebooks;

FIGS. 5A, 5B, 5C and 5D illustrate examples of performance comparisonsfor Rank-2 codebooks;

FIGS. 6A and 6B illustrate examples of performance comparisons forRank-3 codebooks;

FIGS. 7A, 7B, 7C and 7D illustrate examples of performance comparisonswith and without rank adaptation;

FIGS. 8A, 8B, 8C and 8D illustrate additional examples of performancecomparisons with and without rank adaptation;

FIG. 9 illustrates a method of operating a transmitter for use withmultiple transmit antennas as provided by one embodiment of thedisclosure; and

FIG. 10 illustrates a method of operating a receiver as provided by oneembodiment of the disclosure.

DETAILED DESCRIPTION

Embodiments of the present disclosure are given in the context of uplink(UL) communications although the concepts presented in this disclosurealso apply for downlink communications. The embodiments employ atransmitter, which may operate in user equipment (UE) and a receiver,which may operate in a base station (eNodeB/eNB) for a cellular network.The receiver transmits a set of control information to instruct thetransmitter on transmission parameters. Several transmit parameters maybe adapted based on the control information from the receiver. Theseadaptations may include rank (the number of transmission layers)adaptation, spatial transmission stream grouping, link adaptation, andprecoding adaptation.

As noted above, transmission rank is the number of spatial transmissionstreams or layers employed in the transmitter. The transmission rankcannot exceed the number of physical transmit antennas available for useto the transmitter. To reduce control information, it is desirable toreduce the number of spatial transmission streams (or equivalently,codewords) while maintaining the same rank. This can be done by groupingmultiple virtual antennas into one spatial transmission stream. This isessentially a mapping from spatial streams to transmission layers. Linkadaptation employs a modulation and coding scheme (MCS) of each spatialstream. This is determined by the channel quality indication for eachspatial stream.

The current E-UTRA specification (LTE Release 8) supports single-antennatransmission only on an uplink (UL) communication. Hence, all signalingaspects are currently designed with this restriction in mind, and it isbeneficial to improve this aspect of UL communications. Of particularinterest is to increase the UL peak data rate by a factor of four andincrease the UL spectral efficiency to meet IMT-Advanced requirements.Since 64QAM is already supported in LTE Release 8, the support of ULSU-MIMO (including spatial multiplexing) is also desirable.

For downlink (DL) SU-MIMO, codebook-based precoding has proved effectivein achieving desired performance, robustness, and minimizing ofsignaling overhead. The following properties of a precoding codebook forDL are desirable.

-   (1) Property 1: Employing a constant modulus to avoid power    amplifier (PA) imbalance across eNB (base station) antennas. This    means that all non-zero elements of the precoder matrix or vector    have the same magnitude.-   (2) Property 2: Employing a finite alphabet (e.g., QPSK or 8PSK    alphabet) to enable a lower complexity precoder selection at the UE    (user equipment).-   (3) Property 3: A nested property (i.e., any precoder of a lower    rank is a sub-matrix of a precoder of a higher rank) to enable    robust rank override operation at the eNB. This property can also be    used to reduce the precoder search complexity. LTE Release 8    fulfills these properties for DL SU-MIMO codebooks.

For UL SU-MIMO, the following can be inferred based on the differentsystem constraints for UL (as compared to DL).

-   (4) Employing Property 1 above is especially important for UL to    avoid PA imbalance across the UE antennas.-   (5) Employing Property 2 above is desirable to minimize precoder    search complexity. Although an eNB has substantially more computing    power that a UE, the eNB may perform a precoder search for all the    scheduled UEs in a given subframe (TTI) with comparative ease.-   (6) While property 3 above may be beneficial, it is not required for    UL since rank override is not applicable (i.e., rank and precoder    selection at the eNB is performed based on the channel estimate from    the sounding reference signal (SRS)). This is unlike DL, which    relies on RI (rank indication) and PMI (precoder matrix indication)    reports.-   (7) Another important property for UL is to have a low power rating    (e.g., PAPR and cubic metric) for the signal on each transmit    antenna. Together with property 1, this ensures that the UE power    amplifiers operate efficiently by avoiding unnecessary backoff.    Precoder codebook embodiments for UL SU-MIMO based on the above    considerations are described in the following discussions.

FIG. 1 illustrates a transmitter 100 as provided by one embodiment ofthe disclosure. In general, the transmitter 100 employs a plurality P ofphysical antennas whereas up to four physical antennas may be used inexamples discussed below. Using as an example as system compliant to the3GPP LTE-A (Advanced E-UTRA), the transmitter 100 may operate in aDFT-precoded OFDM(A) (also termed DFT-spread OFDM(A)) communicationenvironment although the principles of the present disclosure may beemployed in other communication systems. The transmitter 100 includes atransmission portion 101 and a control information decoding portion 110.The transmission portion 101 includes a modulation, encoding and layermapping unit 102, a precoder unit 104 and a transmit unit 106 havingmultiple OFDM modulators (each typically having IFFT and cyclic prefixinsertion) that feed corresponding transmit antennas. Prior to layermapping, DFT spreading is introduced to spread each of the modulateddata symbols across multiple frequency sub-carriers. The controlinformation decoding portion 110 includes a receive module and a decodermodule.

Generally, the precoder unit 104 is configured to precode data fortransmission using a precoding matrix selected from a codebook 105,wherein the codebook 105 corresponds to the following three transmissionproperties for an uplink transmission: 1) all precoding elements fromthe precoding matrix have a same magnitude, 2) each precoding elementfrom the precoding matrix is taken from a set of finite values, and 3)there is only one non-zero element in any row of the precoding matrix.The transmitter also includes a transmit unit configured to transmit theprecoded data. These codebooks are based on a combination oftransmission properties for an uplink transmission. The transmit unit106 is configured to transmit the precoded data.

In the illustrated embodiment, the control information decoding portion110 recovers a control information transmission from an eNB, which isused to determine a precoding matrix index for each time-frequencyresource block by the precoder unit 104. In addition, themodulation-coding scheme (MCS) and the transmission rank information arealso extracted from the control information. Data are generated andencoded for each symbol stream and transmitted by the transmit unit 106,and the corresponding transmit antennas. The encoded streams are groupedby mapping one spatial stream to multiple spatial layers (in the form ofvirtual antennas) wherein the number of active virtual antennas is givenby a transmission rank R. By performing encoding and layer mapping, thetransmit signal on each transmission layer is assembled.

The precoder unit 104 employs the codebook 105 to convert the signal onthe R virtual antennas to the four physical antennas. Embodiments of theprecoders presented are linear. That is, the signal on each of thephysical antennas is some linear combination of the signals on thevirtual antennas. Thus the mapping can be specified by a linearprecoding matrix. The output of the precoder unit 104 contains signalsto be transmitted on the physical antennas.

In the embodiment shown, the signal is assembled in the frequency domainand converted to the time domain using the OFDM modulators, which canadd a cyclic prefix to guard against channel distortion. The specificDFT-SOFDM modulation is just an example for the various kinds ofmodulation schemes that may be used to send the precoder unit 104outputs on the channel. Other examples may include OFDM(A),code-division multiple access (CDMA) and time-division multiple access(TDMA) transmissions, where the signal is transmitted in the code andtime domains, respectively.

FIG. 2 illustrates a receiver 200 as provided by a correspondingembodiment of the disclosure. The receiver 200 includes a receiveportion 201 and a control information generation portion 210. Thereceive portion 201 includes a receive unit 202, a MIMO detector unit204, a signal processing unit 206, a channel and noise variance unit 207and a precoder selection unit 208. OFDM demodulation in the receive unit202 is followed by channel estimation in the channel and noise varianceunit 207, which enables precoder selection in the precoder selectionunit 208. The precoder selection unit 208 includes a codebook 209.

Generally, the receive unit 202 is configured to receive precoded data,and the precoder selection unit 208 is configured to select a precodingmatrix from the codebook 209 for the received precoded data. Thecodebook 209 provides codebooks corresponding to different transmissionlayers that are based on a combination of transmission properties for anuplink transmission.

The receiver 200 determines, among other things, a transmission rank andprecoder index used by a transmitter (such as the transmitter 100) foreach time-frequency resource. This may be done by decoding one or morecontrol signals, in addition to optionally using the stored codebook209. From the precoder index, the receiver 200 then determines theprecoding matrix used from the codebook 209. This knowledge is used bythe MIMO detector 204 to generate soft information for each transmissionlayer, which is then subsequently processed to decode the various codedstreams.

The control information generation portion 210 uses channel and noisevariance estimates to assemble a set of control information. In thisembodiment, this includes employing a rank selector to provide atransmission rank and the precoder index information from the codebook209 corresponding to precoding selection of the selected transmissionrank. Additionally, modulation-coding scheme (MCS) for the correspondingtransmission layers and coded streams is provided. These quantities arethen sent to the transmitter using a control information encoder.

Different transmission ranks use different precoding codebooks. Toenable the receiver 200 to correctly decode the signal, the precodingmatrix in many communication systems (e.g., E-UTRA) is drawn from afinite codebook. There is a trade-off in the choice of the codebooksize. Increasing the codebook size allows better tuning of the precodermatrix to improve the signal-to-interference plus noise ratio (SINR)seen by the transmission layers, and hence to increase systemthroughput. However, a larger codebook size also increases the amount ofcontrol information signaling to indicate the preferred or actualprecoder index. Also, the eNB may need to compute an optimum precodingmatrix based on channel and noise variance estimates. Referring jointlyto FIGS. 1 and 2, comprehensive examples of codebooks constructedaccording to the principles of the present disclosure are discussedbelow.

Generally, the codebook designs presented fulfill the properties of (1)constant modulus, (2) finite alphabet and (3) a low power rating for thesignal on each transmit antenna. The first two properties are basicallystraightforward to fulfill. Properties (1) and (2) imply that each ofthe precoder elements belongs to an M-PSK constellation. For instance, a4-PSK (QPSK) alphabet implies that the alphabet set is

$\sum\limits_{\;}^{\;}\;{= \{ {{\pm 1},{\pm j}} \}}$while an 8-PSK alphabet implies

$\sum{= {\{ {{\pm 1},{\pm j},\frac{{\pm 1},{\pm j}}{\sqrt{2}}} \}.}}$Note that an additional scaling factor may be needed to ensure thecorrect normalization.

As mentioned, ensuring a low power rating transmission is also desiredin the codebook design. Denoting the number of transmit antennas andspatial layers as P and N, respectively, the transmitted signal can bewritten as shown in equation (1) below.

$\begin{matrix}{{y = { V_{s}\Rightarrow\begin{bmatrix}y_{1} \\y_{2} \\\vdots \\y_{P}\end{bmatrix}  = { {\begin{bmatrix}v_{11} & v_{12} & \ldots & v_{iN} \\v_{21} & v_{22} & \ldots & v_{2N} \\\vdots & \vdots & \ddots & \vdots \\v_{P\; 1} & v_{P\; 2} & \ldots & v_{PN}\end{bmatrix}\begin{bmatrix}s_{1} \\s_{2} \\\vdots \\s_{N}\end{bmatrix}}\Rightarrow y_{P}  = {\sum\limits_{n = 1}^{N}\;{v_{pn}s_{n}}}}}},{{\sum\limits_{n = 1}^{N}\;{v_{pn}}^{2}} = 1.}} & (1)\end{matrix}$Here, s and V denote the transmitted QAM symbols (N-dimensional) and P×Nprecoding matrix, respectively.

Looking at the power of the signal on each transmit antenna and assumingstatistical independence among different transmitted symbols where eachhas a peak power of ρ_(MAX) and an average power of ρ_(AV), equations(2) may be derived.

$\begin{matrix}{{PeakPower} = {{\max{y_{p}}^{2}} = {{{\rho_{MAX}{\sum\limits_{n = 1}^{N}\;{v_{pn}}^{2}}} + {\sum\limits_{n > n^{\prime}}\;{{Re}\{ {v_{pn}^{*}v_{{pn}^{\prime}}s_{n}^{*}s_{n^{\prime}}} \}}}} = {\rho_{MAX}( {1 + {\frac{2}{\rho_{MAX}}{\sum\limits_{n > n^{\prime}}\;{{Re}\{ {v_{pn}^{*}v_{{pn}^{\prime}}s_{n}^{*}s_{n^{\prime}}} \}}}}} )}}}} & (2) \\{\mspace{79mu}{{{AvPower} = {{E{y_{P}}^{2}} = {{\sum\limits_{n = 1}^{N}\;{{v_{pn}}^{2}E{s_{n}}^{2}}} = {{\rho_{AV}{\sum\limits_{n = 1}^{N}\mspace{11mu}{v_{pn}}^{2}}} = \rho_{AV}}}}},\mspace{79mu}{and}}} & \; \\{{PAPR} = {{\frac{\rho_{MAX}}{\rho_{AV}}( {1 + {\frac{2}{\rho_{MAX}}{\sum\limits_{n > n^{\prime}}\;{{Re}\{ {v_{pn}^{*}v_{{pn}^{\prime}}s_{n}^{*}s_{n^{\prime}}} \}}}}} )} = {{PAPR}_{N = 1}( {1 + {\frac{2}{\rho_{MAX}}{\sum\limits_{n > n^{\prime}}\;{{Re}\{ {v_{pn}^{*}v_{{pn}^{\prime}}s_{n}^{*}s_{n^{\prime}}} \}}}}} )}}} & \;\end{matrix}$Note that the condition in equation (1) is automatically satisfied forN=1 since there are only one non-zero term of v_(pn) for a given p.

Now consider a single-stream power rating for 2≦N≦P. It is may be notedfrom equations (2) that the peak-to-average power ratio (PAPR) tends tobe higher compared to N=1 unless the second term in the bracket is zerofor all p=1, 2, . . . , P. Hence, a condition for maintaining the samepower rating as that for a single-stream transmission is shown inequation (3).

$\begin{matrix}{{{\frac{2}{\rho_{MAX}}{\sum\limits_{n > n^{\prime}}\;{{Re}\{ {v_{pn}^{*}v_{{pn}^{\prime}}s_{n}^{*}s_{n^{\prime}}} \}}}} = 0},{p = 1},2,\ldots\mspace{14mu},P} & (3)\end{matrix}$Since the condition in equation (3) has to be valid for all possiblevalues of s_(n)*s_(n′), which may take a large number of values, anecessary and sufficient condition for equation (3) is shown in equation(4).v _(pn) *v _(pn′)=0 for n≠n′, p=1,2, . . . , P  (4)

Hence, the condition of equation (4) allows only one non-zero term ofv_(pn) for a given p for N>1. For P=4, the precoding matrix thatsatisfies this constraint may be written in the form given in equations(5) below.

$\begin{matrix}{{{N = {2\text{:}{\prod_{4 \times 4}\begin{bmatrix}{\exp( {j\;\theta_{1}} )} & 0 \\{\exp( {j\;\theta_{2}} )} & 0 \\0 & {\exp( {j\;\theta_{3}} )} \\0 & {\exp( {j\;\theta_{4}} )}\end{bmatrix}}}},{N = {3\text{:}{\prod_{4 \times 4}\begin{bmatrix}{\exp( {j\;\theta_{1}} )} & 0 & 0 \\0 & {\exp( {j\;\theta_{2}} )} & 0 \\0 & 0 & {\exp( {j\;\theta_{3}} )} \\0 & 0 & {\exp( {j\;\theta_{4}} )}\end{bmatrix}}}},{and}}{N = {4\text{:}{\prod_{4 \times 4}{\begin{bmatrix}{\exp( {j\;\theta_{1}} )} & 0 & 0 & 0 \\0 & {\exp( {j\;\theta_{2}} )} & 0 & 0 \\0 & 0 & {\exp( {j\;\theta_{3}} )} & 0 \\0 & 0 & 0 & {\exp( {j\;\theta_{4}} )}\end{bmatrix}.}}}}} & (5)\end{matrix}$

Here,

∏_(4x4)is a 4×4 permutation or ordering matrix which represents one of the4!=24 possible permutations or orderings. Two examples that representordering (1, 2, 4, 3) and (1, 3, 2, 4) are shown in equation (6) below.

$\begin{matrix}{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\end{bmatrix}} & (6)\end{matrix}$The phase values {θ₁, θ₂, θ₃, θ₄} can be further restricted, forexample, with θ₁=θ₃=0.

Based on the structures in equations (5), a codebook can be designed foreach of the values of N∈{2, 3, . . . , P}. The codebook may be designedto maximize the precoding performance based on one or more criteria suchas maximizing the minimum chordal or the Fubini-Study distance. Notethat the codebook can be designed by combining the structure in equation(5) with some other structures.

Now consider a limited power rating for 2≦N≦P. While the designsconsidered for a single-stream power rating where 2≦N≦P allow a samepower rating as that for N=1 (single-stream precoding), the structuregiven in equation (5) may be too limiting. Furthermore, it may be notedthat the PAs in the UE may not be power-limited for N>1, although insome designs, it is possible to operate the PAs near the back-off regionfor improved efficiency. In this case, a better performing codebook(e.g., having a better distance property) may be obtained by relaxingthe condition of a single-stream power rating.

As an example, the condition in Equation (3) may be modified as follows:

$\begin{matrix}{{{\frac{2}{\rho_{MAX}}{\sum\limits_{n > n^{\prime}}\;{{Re}\{ {v_{pn}^{*}v_{{pn}^{\prime}}s_{n}^{*}s_{n^{\prime}}} \}}}} \leq K},{p = 1},2,\ldots\mspace{14mu},{P.}} & (7)\end{matrix}$Here, K is chosen to limit the PAPR increase over that for N=1.Analogous to the single-stream power rating condition, the condition inequation (7) may be achieved by allowing only a smaller number (Q

N) of non-zero terms of v_(pn) for a given p. For instance, when P=4 andQ=2 there is no particular restriction for N=2. The precoding matrix forN equal to 3 or 4 can be written as follows.

$\begin{matrix}{{{N = {3\text{:}{\prod_{4 \times \; 4}\begin{bmatrix}{\exp( {j\;\theta_{11}} )} & {\exp( {j\;\theta_{12}} )} & 0 \\{\exp( {j\;\theta_{21}} )} & {\exp( {j\;\theta_{22}} )} & 0 \\0 & 0 & {\exp( {j\;\theta_{33}} )} \\0 & 0 & {\exp( {j\;\theta_{43}} )}\end{bmatrix}}}},{and}}{N = {4\text{:}{\prod_{4 \times 4}{\begin{bmatrix}{\exp( {j\;\theta_{11}} )} & {\exp( {j\;\theta_{12}} )} & 0 & 0 \\{{\exp( {j\;\theta_{21}} )}0} & {\exp( {j\;\theta_{22}} )} & 0 & 0 \\0 & 0 & {\exp( {j\;\theta_{33}} )} & {\exp( {j\;\theta_{34}} )} \\0 & 0 & {\exp( {j\;\theta_{43}} )} & {\exp( {j\;\theta_{44}} )}\end{bmatrix}.}}}}} & (8)\end{matrix}$

The parameter definition shown in equations (8) is analogous to thesingle-stream power rating for 2≦N≦P. The example in equations (8) maybe treated as a block diagonal structure where the transmit antennaarray is divided into several antenna subsets (e.g., antenna ports 0 and1 in one subset, and antenna ports 2 and 3 in the other subset, wherethe symbol-to-antenna mapping within one subset is the same and the sizeof each antenna subset is two). Apart from the exemplary design shown inequations (8), it is also possible to adaptively configure the number ofantenna subsets l and the size of the i^(th) subset (L_(i)) such that

${\sum\limits_{i = 1}^{l}\; L_{i}} = {P.}$For instance, one alternative design is shown in equations (9), wherel=4 and L_(i)=1 for i=1, . . . , 4.

$\begin{matrix}{{{N = {3\text{:}{\prod_{4 \times 4}\begin{bmatrix}{\exp( {j\;\theta_{11}} )} & {\exp( {j\;\theta_{12}} )} & 0 \\{\exp( {j\;\theta_{21}} )} & 0 & {\exp( {j\;\theta_{23}} )} \\0 & {\exp( {j\;\theta_{32}} )} & 0 \\0 & 0 & {\exp( {j\;\theta_{43}} )}\end{bmatrix}}}},{and}}{N = {4\text{:}{\prod_{4 \times \; 4}{\begin{bmatrix}{\exp( {j\;\theta_{11}} )} & {\exp( {j\;\theta_{12}} )} & 0 & 0 \\{\exp( {j\;\theta_{21}} )} & 0 & {\exp( {j\;\theta_{23}} )} & 0 \\0 & {\exp( {j\;\theta_{32}} )} & 0 & {\exp( {j\;\theta_{34}} )} \\0 & 0 & {\exp( {j\;\theta_{43}} )} & {\exp( {j\;\theta_{44}} )}\end{bmatrix}.}}}}} & (9)\end{matrix}$

Multi-rank (Rank-2 and Rank-3) precoding may increase the cubic metric(CM) of a transmitted signal. Hence, minimizing a resulting CM isbeneficial for the system. It may be demonstrated that a performancedifference between a CM-preserving (block diagonal) codebook and a DLHouseholder codebook is small for lower transmit correlation, but largerfor higher transmit correlation. At the same time, a reduction in UEtransmit power due to PA backoff needs to be taken into account,especially in power-limited scenarios (e.g., at a cell-edge).

While preserving the CM seems attractive, the following factors are tobe considered. First, imposing a CM-preserving criterion restricts thecodebook structure. This amounts to introducing a number of zerocomponents. This degrades the distance property of the codebook, whichin turn, reduces the potential precoding gain. Second, power limitationis perhaps more relevant for lower-rank transmission (e.g., for UEs onthe cell-edge). In this scenario, a UE is more likely to transmit withpeak power and hence is susceptible to PA backoff. For this scenario, CMpreservation (or more generally, minimizing the CM) is clearlybeneficial. While the “more power is better” principle sounds universal,it is less relevant for higher-rank transmissions. That is, the UEs thatare assigned with higher ranks are less likely to be power limited. Alower rank may be identified with rank-2 while a higher-rank withrank-3, in this case.

From the above discussion, it is also possible to consider thefollowing. For a Rank-2 codebook, impose a CM-preserving criterion andfor a Rank-3 codebook, relax the CM-preserving criterion. TheCM-preserving criterion generally requires an unnecessarily restrictivestructure. However, it is possible to limit the amount of CM increase byutilizing a more specific structure.

More explicit codebook designs for Rank-1, Rank-2, and Rank-3 arepresented below as embodiments of the present disclosure. The codebooksfor different ranks are designed and optimized separately since norequirement to impose artificial relationships across different ranksexists. Concurrently, a total codebook size (Rank-1+Rank-2+Rank-3) notto exceed 63 is maintained in the examples. Clearly, a metric forcodebook design selection is throughput performance. In particular, thefollowing principles may be followed.

Since the codebooks for different ranks are independently designed, arank-n throughput can be used to compare different rank-n codebookdesigns. At the same time, the throughput within a typical geometry(SNR) rank for rank-n may be considered when comparing two designs,which are better in different SNR regions.

Throughput simulation tends to be time-consuming. Therefore, for acodebook search, average mutual information may be used to approximatethe average throughput. A throughput with rank adaptation can be used toevaluate the overall design. Note, however, different rank adaptationalgorithms may yield different conclusions especially since the usagefrequency of different ranks may be quite different for differentalgorithms. Another simple metric used for codebook selection is theminimum chordal distance where the chordal distance between two matricesis defined in equation (10) below.

$\begin{matrix}{{d( {U,V} )} = {{\frac{1}{\sqrt{2}}{{{UU}^{H} - {VV}^{H}}}_{F}\mspace{14mu}{where}\mspace{14mu}{U}_{F}} = \;{{V}_{F} = 1}}} & (10)\end{matrix}$

This metric provides a reasonable indication of the performance in lowspatial correlation. At the same time, this metric may not reflect thethroughput performance for the agreed upon antenna configuration andspatial channel characteristics. For non-uniform antenna configurationssuch as dual-polarized arrays, the antenna element indexing is crucialsince it may heavily affect the performance due to the non-uniformcorrelation profile.

FIGS. 3A, 3B and 3C illustrate an example of transmitter elementindexing for a transmitter and a receiver such as the transmitter 100and receiver 200 shown in FIGS. 1 and 2. FIGS. 3A, 3B and 3C depict theindexing that is assumed in this disclosure. The antenna indexing isused to enumerate the spatial channel coefficients H_(n,m) where n and mare the receiver and transmitter antenna indices, respectively. Observethat the indexing for the two pairs of cross-polarized antennas (FIG.3C) represents the grouping of two antennas with the same polarization,which tend to be more correlated. This is analogous to the indexing oftwo pairs of uniform linear arrays (ULAs) shown in FIG. 3B.

Based on a size constraint of 64 and the need for accommodating severalantenna turn-off vectors, the following codebook sizes are considered.For Rank-1, a size of 16 may be employed without antenna turn-offvectors. For Rank-2, a size of 16 or 20 may be employed. And, forRank-3, a size of 16 may be employed. It is also assumed that a QPSKalphabet is employed, since the gain of an 8PSK alphabet over the QPSKalphabet tends to be marginal. Moreover, a QPSK-based design results ina simpler computational requirement for the precoder search.

For the case of Rank-1 codebooks, several antenna turn-off vectors areto be included in the rank-1 codebook. Additionally, assuming that fourtransmitter (4Tx) UL SU-MIMO is targeted for the antenna configurationshown in FIG. 3C, the following antenna group turn-off vectors shown inequation (11) are suitable along with its permutation. These may bederived from a Rank-1, two transmitter (2Tx) codebook.

$\begin{matrix}{{{\{ {{\frac{1}{2}\begin{bmatrix}1 \\w \\0 \\0\end{bmatrix}},{w \in \{ {{\pm 1},{\pm j}} \}}} \}\bigcup\{ {{\frac{1}{2}\begin{bmatrix}0 \\0 \\1 \\w\end{bmatrix}},{w \in \{ {{\pm 1},{\pm j}} \}}} \}},{or}}{\{ {{\frac{1}{2}\begin{bmatrix}1 \\0 \\w \\0\end{bmatrix}},{w \in \{ {{\pm 1},{\pm j}} \}}} \}\bigcup\{ {{\frac{1}{2}\begin{bmatrix}0 \\1 \\0 \\w\end{bmatrix}},{w \in \{ {{\pm 1},{\pm j}} \}}} \}}} & (11)\end{matrix}$

It may be unnecessary to include all eight vectors. That is, a subset ofsize-4 may be sufficient. Note that the antenna selection vectors aremainly aimed for power saving by turning on only one of four PAs. Hence,the scaling factor of 0.5 is introduced to reflect such an intention.Since the antenna turn-off vectors are typically not chosen in thenormal precoding adaptation unless, for example, severe antenna-gainimbalance occurs.

The constant modulus vectors of the Rank-1 codebook are designedseparately from the antenna turn-off vectors. That is, the antennaturn-off vectors are not included in the minimum chordal distancecomputation and optimization. Also, the antenna turn-off vectors are notsimulated. Hence, the subsequent discussion on the Rank-1 4Tx codebookdesign does not include the antenna turn-off vectors.

To design a Rank-1 codebook with a QPSK alphabet, candidates with thelargest minimum chordal distance are employed. By employing an efficientcodebook search process (an example is provided below), it can be shownthat the four codebooks in Table 1 achieve the largest minimum chordaldistance. It may be observed that the four codebooks are a partition ofthe set of all the 4³=64 possible Rank-1 QPSK precoding vectors. Thefirst, second, and third elements of the n-th precoding vector ofcodebooks 1, 2, 3, and 4 share the same value. The fourth element of then-th precoding vector of codebooks 1, 2, 3, and 4 are related as w_(n)^((k))(4)=(j)^(k−1)w_(n) ^((l))(4), where k indicates the k-th codebook.

Table 1 illustrates Rank-1 codebooks wherein each vector is unit-norm.

TABLE 1 Rank-1 codebooks-not including antenna turn-off vectors Size =16 Minimum chordal distance = 0.866 Mean Chordal distance = 0.893Codebook 1 Index 0 to 7 $\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\j \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\1\end{bmatrix}$ Index 8 to 15 $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\1 \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\{- 1}\end{bmatrix}$ Codebook 2 Index 0 to 7 $\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\j \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\j\end{bmatrix}$ Index 8 to 15 $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\1 \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\{- j}\end{bmatrix}$ Codebook 3 Index 0 to 7 $\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}$ Index 8 to 15 $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\1 \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}$ Codebook 4 Index 0 to 7 $\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\j \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- j}\end{bmatrix}$ Index 8 to 15 $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\1 \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\j\end{bmatrix}$

FIGS. 4A, 4B, 4C and 4D illustrate examples 400, 410, 420 and 430 ofperformance comparisons for Rank-1 codebooks. FIGS. 4A, 4B, 4C and 4Ddepict the comparison of the four codebooks in Table 1 relative to a 4TxDL Householder codebook. Observe that all the codebooks perform equallywell. Codebook 3 in Table 1 performs slightly better than the rest.

While a typical random search process may serve the purpose, it tends tobe inefficient. In a more efficient codebook search process, asmentioned above, a searching mechanism prunes off unwanted matrices fora given starting matrix. The end result depends on a pruning order. Sucha pruning-based algorithm may be described as follows.

-   (1) Generate all the N candidate matrices {W(n), n=1, 2, . . . , N}.    For instance, for a 4Tx Rank-1 codebook employing a QPSK alphabet,    N=64. Index the N matrices as follows:-   (2) Generate the N×N (inter-matrix) chordal distance matrix C.-   (3) Choose a particular chordal distance threshold Th.-   (4) Choose a starting index i_start.-   (5) Look at C(i_start,:) and remove the matrices with a chordal    distance from W(i_start) less than Th. This yields a survivor set.-   (6) Select one of the surviving indices i_surv via randomized or    predetermined ordering (e.g., a trajectory pattern of i_surv    traversing all the survivors). Look at C(i_surv,:) and remove the    matrices with chordal distance from W(i_surv) less than Th. Update    the survivor set.-   (7) Repeat step (6) until all the survivors are exhausted. The final    survivors form a codebook with minimum chordal distance of ≧Th.-   (8) Repeat steps (6) and (7) with different ordering patterns for    i_surv.-   (9) Repeat steps (4) to (7) with different starting indices i_start.-   This procedure can be used to design codebooks for any rank and/or    any size.

For the case of Rank-2 codebooks, a simple design is shown in Table 2.This fits well with the antenna configuration 320 shown in FIG. 3C. Nogroup permutation (i.e., shuffling) is performed across physicalantennas. With a QPSK alphabet, this results in a size-16 codebook. Notethat phase difference across layers does not improve the distanceproperty as well as the SINR per layer. Hence, the first non-zerocoefficient associated with each layer is zero-phased. It is, of course,possible to use any alphabet size such as an 8PSK. The gain, however, isexpected to be marginal if any.

Table 2 illustrates a Rank-2 codebook, where each matrix is unit-norm,and shows a simple structure that is designed based on a XPD structure.

TABLE 2 Rank-2 codebook A Simple design without antenna grouppermutation Size = 16 Minimum chordal distance = 0.3536 Mean chordaldistance = 0.5047 ${w = {\frac{1}{2}\begin{bmatrix}1 & 0 \\w_{1} & 0 \\0 & 1 \\0 & w_{2}\end{bmatrix}}},{w_{n} \in \{ {{\pm 1},{\pm j}} \}}$

Another design may be obtained by incorporating antenna grouppermutation. There are six possible antenna group permutations relativeto the spatial channel. The permutations are {(1, 2, 3, 4), (1, 3, 2,4), (1, 4, 2, 3), (2, 3, 1, 4), (2, 4, 1, 3)(3, 4, 1, 2)}. Notice thatthe last three permutations can be derived from the first three byswapping the layers. However, it may also be seen that the chordaldistance is invariant to permutation or swapping across the layers.

$\begin{matrix}{{{{{U{\prod_{1}{\prod_{1}^{H}U^{H}}}} - {V{\prod_{2}{\prod_{2}^{H}V^{H}}}}}}_{F} = {{{UU}^{H} - {VV}^{H}}}_{F}}{{{since}\mspace{14mu}{\prod_{1}\prod_{1}^{H}}} = I}} & (12)\end{matrix}$

Such an invariance property also holds for SINR. Hence, only threepermutations need to be considered for the codebook design. They are{(1, 2, 3, 4), (1, 3, 2, 4), (1, 4, 2, 3)}. In this case, all the 48possible matrices can be written as a union of the three matrix sets asshown in equation (13).

$\begin{matrix}{\{ {{\frac{1}{2}\begin{bmatrix}1 & 0 \\w_{1} & 0 \\0 & 1 \\0 & w_{2}\end{bmatrix}},{w_{n} \in \{ {{\pm 1},{\pm j}} \}}} \}\bigcup\{ {{\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\w_{1} & 0 \\0 & w_{2}\end{bmatrix}},{w_{n} \in \{ {{\pm 1},{\pm j}} \}}} \}\bigcup\{ {{\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & w_{2} \\w_{1} & 0\end{bmatrix}},{w_{n} \in \{ {{\pm 1},{\pm j}} \}}} \}} & (13)\end{matrix}$The first set corresponds to the permutation (1, 2, 3, 4) while thesecond and the third correspond to (1, 3, 2, 4) and (1, 4, 2, 3),respectively.

While a random search may provide a good codebook, a codebook can bedesigned by considering the underlying precoder structure in relation tothe antenna configuration as shown in FIG. 3C. The followingobservations can be inferred from equation (13). The chordal distancebetween two matrices within one set (of the three sets) can be writtenas equation (14).

$\begin{matrix}{{d( {U,V} )} = {\frac{1}{4}\sqrt{{{u_{1} - v_{1}^{2}}} + {{u_{2} - v_{2}}}^{2}}}} & (14)\end{matrix}$

As given in Table 2, the minimum chordal distance of the first set (ofsize-16) is 0.3538. From equation (14), it can be shown that there aretwo non-intersecting subsets of size-8 with a minimum and mean chordaldistance of 0.5 and 0.5296, respectively. Also note that the minimumchordal distance of a smaller subset with more than one matrix cannotexceed 0.5, as will be shown in Table 3. This also holds for the othertwo sets in equation (13). The same more efficient codebook searchprocess example described above may be employed.

The chordal distance between any matrix from one of the three sets andany other matrix from another set is 0.5. This implies the following.Two other size-8 subsets can be derived from one of the subsets in Table3 by applying antenna group permutations (1, 3, 2, 4) and (1, 4, 2, 3).The three size-8 subsets can be combined to form a size-24 codebook witha minimum chordal distance of 0.5.

In total, four other size-8 subsets can be derived from subsets 1-1 and1-2. They are 2-1, 2-2, 3-1, and 3-2. Here, the designation 1-1indicates the first permutation (1, 2, 3, 4) corresponding to the firstsubset in Table 3. For x=1 or 2, subset 2-x is derived by multiplyingeach of the matrices in subset 1-x with

$\quad\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\end{bmatrix}$(associated with the permutation (1, 3, 2, 4)). Similarly, subset 3-x isderived from subset 1-x using

$\quad\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0\end{bmatrix}$(associated with the permutation (1, 4, 2, 3)).

Furthermore, any size-24 combination across the three size-8 subsetscorresponding to different antenna group permutation also results in acodebook with a minimum chordal distance of 0.5. For instance, thecombination of 1-1, 2-2, and 3-1 results in a codebook with a minimumchordal distance of 0.5. There are a total of 2³=8 combinations.Consequently, a size-16 or size-20 codebook with a minimum chordaldistance of 0.5 can be obtained by pruning any of the eight possiblesize-24 codebooks mentioned above. Two examples are given in Table 4where the size-16 codebook is composed of subset 1-2 (eight matrices),four matrices from subset 2-1, and four matrices from subset 3-2. Thesize-20 codebook is composed of subset 1-1 (matrices), four matricesfrom subset 2-1 and subset 3-1 for a total of eight matrices). It mayalso be noted that expanding the alphabet from a QPSK to an 8PSK doesnot improve the distance property.

Table 3 illustrates two non-intersecting subsets of set 1 in equation(13) with a minimum chordal distance of 0.5. Table 4 illustrates Rank-2codebooks, where each matrix is unit-norm.

TABLE 3 Subset 1-1: permutation (1, 2, 3, 4) $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ Subset 1-2: permutation (1, 2, 3, 4)$\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$

TABLE 4 Rank-2 codebook B.1 Allow antenna grouping permutation Size = 16Minimum chordal distance = 0.500 Mean chordal distance = 0.533 Index 0to 6 $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ Index 7 to 13 $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix}$ Index 14 to 15 $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- 1} & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix}$ Rank-2 codebook B.2 Allow antenna grouping permutationSize = 20 Minimum chordal distance = 0.500 Mean chordal distance = 0.526Index 0 to 6 $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & j\end{bmatrix}$ Index 7 to 13 $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\1 & 0\end{bmatrix}$ Index 14 to 19 $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- 1} & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\j & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- j} & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix}$

FIGS. 5A, 5B, 5C and 5D illustrate examples 500, 510, 520 and 530 ofperformance comparisons for Rank-2 codebooks. The examples 500, 510, 520and 530 depict throughput comparisons of the following Rank-2 codebooks.

-   (1) A 4Tx Householder codebook for DL MIMO Release 8.-   (2) A CM-preserving codebook A (Table 2) without antenna group    permutation for codebook size-16.-   (3) A CM-preserving codebook B.1 (Table 3) without antenna group    permutation for codebook size-16.-   (4) A CM-preserving codebook B.2 (Table 3) without antenna group    permutation for codebook size-20.

The rank is fixed to two for this comparison. The gain relative to theHouseholder codebook is also shown. The following observations may bemade. Codebooks B-1 (size-16) and B-2 (size-20) perform approximatelythe same. Codebooks B-1 and B-2 offer significant gain over codebook A.Therefore, antenna group permutation is seen to be beneficial.

For the case of Rank-3 codebooks, a CM-preserving design is consideredfirst. A Rank-3 CM-preserving design may be written as follows.

$\begin{matrix}{W = {\frac{1}{2} \times {\prod{\quad{\begin{bmatrix}1 & 0 & 0 \\w & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}.}}}}} & (15)\end{matrix}$The coefficient w belongs to an alphabet set. For instance, w∈{±1,±j}holds for a QPSK-based design. Other alphabet sets such as 8PSK

$( {w \in \{ {{\pm 1},{\pm j},\frac{{\pm 1},{\pm j}}{\sqrt{2}}} \}} )$are also possible. Similar to the Rank-2 design, a codebook distanceproperty and SINR are invariant to the phase difference across layers.Hence, the first non-zero coefficient associated with each layer iszero-phased.

The 4×4 matrix Π represents permutation or ordering operation. Whilethere are a total of 4!=24 possible orderings, the following facts canbe used to reduce the number of permutation matrices we need forcodebook design.

-   (1) As for the Rank-2 design, the distance property and SINR are    invariant to permutation across layers.-   (2) Taking into account the layer mapping for the 3-layer    transmission, only the first layer may be assigned with two non-zero    coefficients (which corresponds to mapping onto two physical    antennas). This is because the first codeword is mapped onto one    layer while the second codeword is mapped onto two layers. Note that    the first layer is assigned 2× power compared to the other layers.

Based on the above facts, only six permutations are considered. The setof all possible candidates can be written as shown below.

$\begin{matrix}\begin{Bmatrix}{{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\w & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\w & 0 & 0 \\0 & 0 & 1\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\w & 0 & 1\end{bmatrix}},} \\{{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\w & 1 & 0 \\0 & 0 & 1\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 0 \\w & 0 & 1\end{bmatrix}},{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\w & 0 & 0\end{bmatrix}}}\end{Bmatrix} & (16)\end{matrix}$

Each of the six permutation groups is termed a permutation set. Theabove structure can be used to optimize the codebook design. The chordaldistance between two matrices within one permutation set (of the sixsets) can be written as:

$\begin{matrix}{{d( {U,V} )} = {\frac{1}{4}{{u - v}}}} & (17)\end{matrix}$Hence, the codebook distance property is limited by the minimum distanceof the alphabet set. In this case, increasing the alphabet size fromQPSK and 8PSK does not improve the codebook distance property for agiven codebook size. With a QPSK alphabet,

${d_{\min}( {U,V} )} = {{\frac{1}{4}\sqrt{2}} = {\frac{1}{2\sqrt{2}}.}}$The chordal distance between any matrix from one of the six sets and anyother matrix from another set is

$\frac{1}{2\sqrt{2}}.$

Assuming a QPSK alphabet, any subset of the size-24 set specified inequation (16) forms a codebook with a minimum chordal distance of

$\frac{1}{2\sqrt{2}}.$The following can be implied if a size-16 codebook is desired.

-   (1) There are a total of

$\quad\begin{pmatrix}24 \\16\end{pmatrix}$possibilities that guarantee the largest minimum chordal distance. Butany of such possibilities does not necessarily maximize the mean chordaldistance.

-   (2) A Design A is a size-16 codebook having good mean chordal    distance accomplished by taking only any 4 out of 6 permutation    groups. There are

$\begin{pmatrix}6 \\4\end{pmatrix} = 15$of such possibilities although such chordal distance is not maximum.(3) A Design B is another size-16 codebook having the same mean chordaldistance as Design A can be obtained as follows.

(a) For 12 matrices, take two matrices with the largest distance fromeach of the six permutation groups. This can be easily seen fromequations (16) and (17). That is (u,v)=(+1,−1) or (u,v)=(+j,−j), whichresults in

${d( {U,V} )} = {\frac{1}{2}.}$Assume that (u,v)=(+1,−1) for all the six permutation groups. Otherwisethere are a total of 2⁶=64 possible combinations of the two choicesacross the 6 permutation groups.

(b) For four matrices: Take any four matrices from the remaining 12matrices.

There are

$\begin{pmatrix}12 \\4\end{pmatrix} = 395$possibilities. However, the number of candidates can be reduced if twoof the four matrices are taken from one permutation group (to complementthe two previously chosen matrices) and the other two from one otherpermutation group. In this case, there are

$\begin{pmatrix}6 \\2\end{pmatrix} = 15$possibilities.

With this construction, all the permutation sets are used. In the two ofthe six permutation groups all four possible precoding matrices areused. In the other four permutation groups, only two matrices are usedas indicated above. Two examples of the design are given in Table 5.

Table 5 illustrates Rank-3 CM-preserving codebooks where each matrix isunit-norm.

TABLE 5 Rank-3 codebook A Permutation groups: P1, P3, P4, P6 Size = 16Minimum chordal distance = 0.3538 Mean chordal distance = 0.3633 Index 0to 3 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 4 to 7 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}$ Index 8 to 11 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 12 to 15 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\j & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- j} & 0 & 0\end{bmatrix}$ Rank-3 codebook B Permutation groups: P1, P2, P3, P6 Size= 16 Minimum chordal distance = 0.3538 Mean chordal distance = 0.3633Index 0 to 5 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ Index 6 to 11 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$ Index 12 to 15 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}$ Rank-3 codebook C Permutation groups: P1, P3, P4, P6 Size= 16 Minimum chordal distance = 0.3538 Mean chordal distance = 0.3633Index to 4 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix}$ Index 6 to 11 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 12 to 15 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\j & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- j} & 0 & 0\end{bmatrix}$

FIGS. 6A and 6B illustrate examples of performance comparisons 600, 610for Rank-3 codebooks. The 4Tx DL Householder codebook may be used as areference. Again, the Householder codebook performs better thanCM-preserving designs. Overall, Design A given in Table 5 offers thebest performance among the CM-preserving designs.

It is also possible to design larger codebooks, for example, size-20.Using the approach in Design A, there are a total of six possibilities:Alt1 (P1,P2,P3.P4,P5), Alt2 (P1,P2,P3,P4,P6), Alt3 (P1,P2,P3,P5,P6),Alt4 (P1,P2,P4,P5,P6), Alt5 (P1,P3,P4,P5,P6), and Alt6 (P2,P3,P4,P5,P6).It may be observed that alternative 4 (Alt4), as shown in Table 6,appears to perform best overall as compared in FIGS. 7A through 7D.

Table 6 illustrates Rank-3 CM-preserving codebooks where each matrix isunit-norm for a size-20 example.

TABLE 6 Rank-3 codebook A Permutation groups: P1, P2, P4, P5, P6 (Alt4)Size = 20 Minimum chordal distance = 0.3538 Mean chordal distance =0.3633 Index 0 to 3 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 4 to 7 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 8 to 11 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 12 to 15 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}$ Index 16 to 19 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\j & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- j} & 0 & 0\end{bmatrix}$

It is also possible to design smaller codebooks, for example, size-12.Possible alternatives may be stated as follows.

-   (1) For Alt1, use all the six permutation groups and a BPSK-only    alphabet.-   (2) For Alt2, select only three out of six permutation groups. For    each permutation group, a QPSK alphabet is used.-   (3) For Alt3, select two permutation groups where a QPSK alphabet is    used and use the remaining four permutations with only a size-1    alphabet.    Design examples for the above alternatives are given in Table 7.

Table 7 illustrates Rank-3 CM-preserving codebook designs (size-12).

TABLE 7 Rank-3 codebook E Alt1 Permutation groups: P1, P2, P3, P4, P5,P6 BPSK alphabet Size = 12 Minimum chordal distance = 0.3538 Meanchordal distance = 0.3716 Index 0 to 3 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 4 to 7 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 8 to 11 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$ Rank-3 codebook E Alt2 Permutation groups: P1, P4, P6QPSK alphabet Size = 12 Minimum chordal distance = 0.3538 Mean chordaldistance = 0.3716 Index 0 to 3 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 4 to 7 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 8 to 11 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\j & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- j} & 0 & 0\end{bmatrix}$ Rank-3 codebook E Alt3 QPSK alphabet with P1, P6;Single-alphabet with P2, P3, P4, P5 Size = 12 Minimum chordal distance =0.3538 Mean chordal distance = 0.3716 Index 0 to 3$\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ Index 4 to 7 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ Index 8 to 11 $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\j & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- j} & 0 & 0\end{bmatrix}$

FIGS. 8A, 8B, 8C and 8D illustrate additional examples of performancecomparisons with and without rank adaptation. Additional codebookdesigns are also simulated as references.

FIG. 9 illustrates a flow diagram of a method 900 of operating atransmitter as provided by one embodiment of the disclosure. The methodstarts in a step 905, and a transmitter having multiple transmitantennas is provided in a step 910. Then, a precoding matrix is selectedfrom a codebook, wherein the codebook corresponds to the following threetransmission properties for an uplink transmission: 1) all precodingelements from the precoding matrix have a same magnitude, 2) eachprecoding element from the precoding matrix is taken from a set offinite values and 3) there is only one non-zero element in any row ofthe precoding matrix in a step 915.

In one embodiment, the codebook for a rank-1 transmission corresponds toa set of four-element column vectors containing at least a subset of thefollowing sixteen vectors shown in Table 8, below.

TABLE 8 $\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\1 \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}$

In another embodiment, the codebook for a rank-2 transmissioncorresponds to a set of four-by-two matrices with one column having twonon-zero elements in two adjacent rows, a remaining column having twonon-zero elements in two remaining adjacent rows and, zero elements inall remaining element positions and contains at least a subset of thefollowing eight matrices shown in Table 9, below.

TABLE 9 $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$

In yet another embodiment, the codebook for a rank-3 transmissioncorresponds to a set of four-by-three matrices with a first columnhaving two non-zero elements, a second column having one non-zeroelement, a third column having one non-zero element, zero elements inall remaining element positions and contains at least a subset of thefollowing twelve matrices shown in Table 10, below.

TABLE 10 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$The precoded data is transmitted in a step 920 and the method 900 endsin a step 925.

FIG. 10 illustrates a flow diagram of a method of operating a receiveras provided by one embodiment of the disclosure. The method 10 starts ina step 1005, and a receiver is provided in a step 1010. Then, precodeddata from a transmitter having multiple transmit antennas is received ina step 1015.

A precoding matrix is selected from a codebook in a step 1020, whereinthe codebook corresponds to the following three transmission propertiesfor an uplink transmission: 1) all precoding elements from the precodingmatrix have a same magnitude, 2) each precoding element from theprecoding matrix is taken from a set of finite values and 3) there isonly one non-zero element in any row of the precoding matrix.

In one embodiment, the codebook for a rank-1 transmission corresponds toa set of four-element column vectors containing at least a subset of thefollowing sixteen vectors shown in Table 11, below.

TABLE 11 $\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\1 \\1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{–1} \\j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}$

In another embodiment, the codebook for a rank-2 transmissioncorresponds to a set of four-by-two matrices with one column having twonon-zero elements in two adjacent rows, a remaining column having twonon-zero elements in two remaining adjacent rows and, zero elements inall remaining element positions and contains at least a subset of thefollowing eight matrices shown in Table 12, below.

TABLE 12 $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}$

In yet another embodiment, the codebook for a rank-3 transmissioncorresponds to a set of four-by-three matrices with a first columnhaving two non-zero elements, a second column having one non-zeroelement, a third column having one non-zero element, zero elements inall remaining element positions and contains at least a subset of thefollowing twelve matrices shown in Table 13, below.

TABLE 13 $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}$ $\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}$The method 900 ends in a step 925.

While the methods disclosed herein have been described and shown withreference to particular steps performed in a particular order, it willbe understood that these steps may be combined, subdivided, or reorderedto form an equivalent method without departing from the teachings of thepresent disclosure. Accordingly, unless specifically indicated herein,the order or the grouping of the steps is not a limitation of thepresent disclosure.

Those skilled in the art to which the disclosure relates will appreciatethat other and further additions, deletions, substitutions andmodifications may be made to the described example embodiments withoutdeparting from the disclosure.

What is claimed is:
 1. A transmitter for use with multiple transmitantennas, comprising: a precoder unit configured to precode data for atransmission using a precoding matrix selected from a codebook, whereinthe codebook corresponds to the following transmission properties for anuplink transmission: 1) all precoding elements from the precoding matrixhave a same magnitude, 2) each precoding element from the precodingmatrix is taken from a set of finite values; and a transmit unitconfigured to transmit the precoded data, wherein the codebook for arank-1 transmission corresponds to a set of four-element column vectorscontaining all of the following sixteen vectors:${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}{\mspace{11mu} 1} \\{\mspace{11mu} 1} \\{- 1} \\{\mspace{11mu} 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 2. A transmitter for use with multiple transmitantennas, comprising: a precoder unit configured to precode data for atransmission using a precoding matrix selected from a codebook, whereinthe codebook corresponds to the following transmission properties for anuplink transmission: 1) all precoding elements from the precoding matrixhave a same magnitude, 2) each precoding element from the precodingmatrix is taken from a set of finite values; and a transmit unitconfigured to transmit the precoded data, wherein the codebook for arank-2 transmission corresponds to a set of four-by-two matrices withone column having two non-zero elements in two adjacent rows, aremaining column having two non-zero elements in two remaining adjacentrows and zero elements in all remaining element positions, and whereinthe codebook for a rank-1 transmission corresponds to a set offour-element column vectors containing all of the following sixteenvectors: ${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}}\;$
 3. The transmitter as recited in claim 2 whereinthe codebook for the rank-2 transmission contains at least a subset ofthe following eight matrices: ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{matrix} \rbrack}$ ${\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}}.}$
 4. The transmitter as recited in claim 2 wherein thecodebook for the rank-2 transmission contains all of the following eightmatrices: ${\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}}.}$
 5. A transmitter for use with multiple transmitantennas, comprising: a precoder unit configured to precode data for atransmission using a precoding matrix selected from a codebook, whereinthe codebook corresponds to the following transmission properties for anuplink transmission: 1) all precoding elements from the precoding matrixhave a same magnitude, 2) each precoding element from the precodingmatrix is taken from a set of finite values; and a transmit unitconfigured to transmit the precoded data, wherein the codebook for arank-3 transmission corresponds to a set of four-by-three matrices witha first column having two non-zero elements, a second column having onenon-zero element, a third column having one non-zero element and zeroelements in all remaining element positions, and wherein the codebookfor a rank-1 transmission corresponds to a set of four-element columnvectors containing all of the following sixteen vectors:${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 6. The transmitter as recited in claim 5 wherein thecodebook for the rank-3 transmission contains at least a subset of thefollowing twelve matrices: ${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}}.}$
 7. The transmitter as recited in claim 5 wherein thecodebook for the rank-3 transmission contains all of the followingtwelve matrices: ${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}}.}$
 8. A method of operating a transmitter for use withmultiple transmit antennas, comprising: selecting a precoding matrixfrom a codebook, wherein the codebook corresponds to the followingtransmission properties for an uplink transmission: 1) all precodingelements from the precoding matrix have a same magnitude, 2) eachprecoding element from the precoding matrix is taken from a set offinite values; and transmitting, by the transmitter, the precoded data,wherein the codebook for a rank-1 transmission corresponds to a set offour-element column vectors containing all of the following sixteenvectors: ${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 9. A method of operating a transmitter for use withmultiple transmit antennas, comprising: selecting a precoding matrixfrom a codebook, wherein the codebook corresponds to the followingtransmission properties for an uplink transmission: 1) all precodingelements from the precoding matrix have a same magnitude, 2) eachprecoding element from the precoding matrix is taken from a set offinite value; and transmitting, by the transmitter, the precoded data,wherein the codebook for a rank-2 transmission corresponds to a set offour-by-two matrices with one column having two non-zero elements in twoadjacent rows, a remaining column having two non-zero elements in tworemaining adjacent rows and, zero elements in all remaining elementpositions and contains at least a subset of the following eightmatrices: ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{matrix} \rbrack}$ ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{matrix} \rbrack}$ and wherein the codebook for a rank-1transmission corresponds to a set of four-element column vectorscontaining all of the following sixteen vectors:${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 10. The method as recited in claim 9, wherein thecodebook for a rank-3 transmission corresponds to a set of four-by-threematrices with a first column having two non-zero elements, a secondcolumn having one non-zero element, a third column having one non-zeroelement, zero elements in all remaining element positions and containsat least a subset of the following twelve matrices:${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}}.}$
 11. The method of claim 9, wherein zero elements inall remaining element positions and contains all of the following eightmatrices: ${\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}}.}$
 12. A receiver, comprising: a receive unit configuredto receive precoded data from a transmitter having multiple transmitantennas; and a precoder selection unit configured to select a precodingmatrix from a codebook, wherein the codebook corresponds to thefollowing transmission properties for an uplink transmission: 1) allprecoding elements from the precoding matrix have a same magnitude, 2)each precoding element from the precoding matrix is taken from a set offinite values, wherein the codebook for a rank-1 transmissioncorresponds to a set of four-element column vectors containing all ofthe following sixteen vectors: ${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 13. A receiver, comprising: a receive unit configuredto receive precoded data from a transmitter having multiple transmitantennas; and a precoder selection unit configured to select a precodingmatrix from a codebook, wherein the codebook corresponds to thefollowing transmission properties for an uplink transmission: 1) allprecoding elements from the precoding matrix have a same magnitude, 2)each precoding element from the precoding matrix is taken from a set offinite values, wherein the codebook for a rank-2 transmissioncorresponds to a set of four-by-two matrices with one column having twonon-zero elements in two adjacent rows, a remaining column having twonon-zero elements in two remaining adjacent rows and zero elements inall remaining element positions, and wherein the codebook for a rank-1transmission corresponds to a set of four-element column vectorscontaining all of the following sixteen vectors:${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 14. The receiver as recited in claim 13 wherein thecodebook for the rank-2 transmission contains at least a subset of thefollowing eight matrices: ${\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}}.}$
 15. The receiver of claim 13, wherein the codebook forthe rank-3 transmission contains all of the following twelve matrices:${\frac{1}{2}\lbrack \begin{matrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0 \\{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 1} & 0 & 0 \\{\mspace{14mu} 0} & 1 & 0 \\{- 1} & 0 & 0 \\{\mspace{14mu} 0} & 0 & 1\end{matrix} \rbrack}$ ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 1} & 0 & 0 \\{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{- 1} & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0 \\{\mspace{14mu} 0} & 0 & 1\end{matrix} \rbrack}$ ${\frac{1}{2}\lbrack \begin{matrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 1} & 0 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{- 1} & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0\end{matrix} \rbrack}.}$
 16. A receiver, comprising: a receiveunit configured to receive precoded data from a transmitter havingmultiple transmit antennas; and a precoder selection unit configured toselect a precoding matrix from a codebook, wherein the codebookcorresponds to the following transmission properties for an uplinktransmission: 1) all precoding elements from the precoding matrix have asame magnitude, 2) each precoding element from the precoding matrix istaken from a set of finite values, wherein the codebook for a rank-3transmission corresponds to a set of four-by-three matrices with a firstcolumn having two non-zero elements, a second column having one non-zeroelement, a third column having one non-zero element and zero elements inall remaining element positions, and wherein the codebook for a rank-1transmission corresponds to a set of four-element column vectorscontaining all of the following sixteen vectors:${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 17. The receiver as recited in claim 16 wherein thecodebook for the rank-3 transmission contains at least a subset of thefollowing twelve matrices: ${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}}.}$
 18. The receiver of claim 16, wherein the codebook forthe rank-3 transmission contains all of the following twelve matrices:${\frac{1}{2}\lbrack \begin{matrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0 \\{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 1} & 0 & 0 \\{\mspace{14mu} 0} & 1 & 0 \\{- 1} & 0 & 0 \\{\mspace{14mu} 0} & 0 & 1\end{matrix} \rbrack}$ ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 1} & 0 & 0 \\{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{- 1} & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0 \\{\mspace{14mu} 0} & 0 & 1\end{matrix} \rbrack}$ ${\frac{1}{2}\lbrack \begin{matrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 1} & 0 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{- 1} & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0\end{matrix} \rbrack}.}$
 19. A method of operating a receiver,comprising: receiving, by the receiver, precoded data from a transmitterhaving multiple transmit antennas; and selecting a precoding matrix froma codebook, wherein the codebook corresponds to the followingtransmission properties for an uplink transmission: 1) all precodingelements from the precoding matrix have a same magnitude, 2) eachprecoding element from the precoding matrix is taken from a set offinite values, wherein the codebook for a rank-1 transmissioncorresponds to a set of four-element column vectors containing all ofthe following sixteen vectors: ${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 20. The method of claim 19, further comprising zeroelements in all remaining element positions and contains all of thefollowing twelve matrices: ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0 \\{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{matrix} \rbrack}$ ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 1} & 0 & 0 \\{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{- 1} & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0 \\{\mspace{14mu} 0} & 0 & 1\end{matrix} \rbrack}$ ${\frac{1}{2}\lbrack \begin{matrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 1} & 0 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{- 1} & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{\; 2}\lbrack \begin{matrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{matrix} \rbrack}\mspace{14mu}{{\frac{1}{\; 2}\lbrack \begin{matrix}{\mspace{14mu} 0} & 1 & 0 \\{\mspace{14mu} 0} & 0 & 1 \\{\mspace{14mu} 1} & 0 & 0 \\{- 1} & 0 & 0\end{matrix} \rbrack}.}$
 21. A method of operating a receiver,comprising: receiving, by the receiver, precoded data from a transmitterhaving multiple transmit antennas; and selecting a precoding matrix froma codebook, wherein the codebook corresponds to the followingtransmission properties for an uplink transmission: 1) all precodingelements from the precoding matrix have a same magnitude, 2) eachprecoding element from the precoding matrix is taken from a set offinite values, wherein the codebook for a rank-2 transmissioncorresponds to a set of four-by-two matrices with one column having twonon-zero elements in two adjacent rows, a remaining column having twonon-zero elements in two remaining adjacent rows and, zero elements inall remaining element positions and contains at least a subset of thefollowing eight matrices: ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{matrix} \rbrack}$ ${\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{matrix} \rbrack}\mspace{14mu}{\frac{1}{2}\lbrack \begin{matrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{matrix} \rbrack}$ and wherein the codebook for a rank-1transmission corresponds to a set of four-element column vectorscontaining all of the following sixteen vectors:${{\frac{1}{2}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\j \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\1 \\{- j} \\{- j}\end{bmatrix}}}\mspace{14mu}$ ${\frac{1}{2}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\j \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\j \\{- j} \\{- 1}\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}\begin{matrix}1 \\{- 1}\end{matrix} \\1 \\1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\j \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- 1} \\{- j} \\j\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\j \\{- 1}\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}1 \\{- j} \\{- j} \\1\end{bmatrix}}.}$
 22. The method as recited in claim 21 wherein thecodebook for a rank-3 transmission corresponds to a set of four-by-threematrices with a first column having two non-zero elements, a secondcolumn having one non-zero element, a third column having one non-zeroelement, zero elements in all remaining element positions and containsat least a subset of the following twelve matrices:${{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}}\;$ ${\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix}}$ ${\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}}\mspace{14mu}{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix}}\mspace{14mu}{{\frac{1}{2}\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix}}.}$